Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 113 / 1 / c / Solution

Let $X={\bigcup }_{i=1}^r{U}_i$ be the stated finite affine cover. By the description of sections on a principal open subscheme, after increasing denominators separately we may writefor some $a_i\in \Gamma (U_i,{\mathcal{O}}_X)$. Multiplying the numerators by powers of $f$ lets us use one exponent $n$ for every $i$.

On $W_{ij}=U_i\cap U_j$, the section $a_i-a_j$ vanishes after restriction to $(W_{ij}{)}_f$. Each $W_{ij}$ is quasi-compact, so part (b) supplies $m_{ij}$ with $f^{m_{ij}}(a_i-a_j)=0$ on $W_{ij}$. Choose one $m$ valid for all the finitely many pairs. The sections $f^m{a}_i$ now agree on every overlap, and the gluing axiom of a sheaf of rings produces $a\in \Gamma (X,{\mathcal{O}}_X)$ with $a{|}_{U_i}=f^m{a}_i$. On $X_f$,Thus some power of $f$ times $b$ extends to a global regular function on $X$.